Equivalence relation

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Definition

A relation [ilmath]\sim[/ilmath] in [ilmath]X[/ilmath] is an equivalence relation if it has the following properties:

• Reflexivity, [ilmath]x\sim x[/ilmath]
• Symmetricity, [ilmath]x\sim y[/ilmath] implies [ilmath]y\sim x[/ilmath]
• Transitivity, [ilmath]x\sim y[/ilmath] and [ilmath]y\sim z[/ilmath] implies [ilmath]x\sim z[/ilmath].

	Name	Definition
1	Reflexive	[ilmath]\forall x\in X[(x,x) \in \sim][/ilmath]. Often written [ilmath]\forall x\in X[x\sim x][/ilmath].
2	Symmetric	[ilmath]\forall x,y\in X[M[/ilmath]. Often written [ilmath]\forall x,y \in X[x\sim y \implies y\sim x][/ilmath].
3	Transitive	[ilmath]\forall x,y,z\in X[((x,y) \in \sim \wedge (y,z) \in \sim) \implies (x,z) \in \sim][/ilmath]. Often written [ilmath]\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z][/ilmath].

Terminology

• Sometimes, letters and other designations are used with symbols to distinguish between different equivalence relations, such as [ilmath]a \equiv_x b[/ilmath].
  • For an [ilmath]x\in X[/ilmath], the equivalence class is written [ilmath][x][/ilmath] or [ilmath]x_\sim[/ilmath]. That is, [ilmath]\forall a\in X[a\in[x] \implies a\sim x][/ilmath].

See Also

• Relation
• Equivalence class
  • An equivalence class partitions a set.

Old Page

An equivalence relation is a special kind of relation

Required properties

Given a relation [ilmath]R[/ilmath] in [ilmath]A[/ilmath] we require the following properties to define a relation (these are restated for convenience from the relation page)

Reflexive

A relation [ilmath]R[/ilmath] if for all [ilmath]a\in A[/ilmath] we have [ilmath]aRa[/ilmath]

Symmetric

A relation [ilmath]R[/ilmath] is symmetric if for all [ilmath]a,b\in A[/ilmath] we have [ilmath]aRb\implies bRa[/ilmath]

Transitive

A relation [ilmath]R[/ilmath] is transitive if for all [ilmath]a,b,c\in A[/ilmath] we have [ilmath]aRb\text{ and }bRc\implies aRc[/ilmath]

Definition

A relation [ilmath]R[/ilmath] is an equivalence relation if it is:

• reflexive
• symmetric
• transitive